Examples on Circumcircles Incircles and Excircles Set 6

Tangents are drawn to the incircle of \(\Delta ABC\) which are parallel to its sides. Let \(x,\;y,\;z\) be the lengths of the tangents opposite \(a,\;b,\;c\) to respectively. Find the value of \(\begin{align}\frac{x}{a} + \frac{y}{b} + \frac{z}{c}\end{align}\) .

Solution:

Now, observe that the incircle of \(\Delta ABC\) is the ex-circle opposite to A for \(\Delta ADE\) . The semiperimeter of \(\Delta ADE\) is

In a \(\Delta ABC\) , \(a,\;c,\;\angle A\) are fixed. The third side may have two possible value say \({b_1}\) and \({b_2}\) . It is given that \({b_2} = 2{b_1}\) . Find the value of \(\begin{align}\frac{c}{a}\sqrt {1 + 8{{\sin }^2}A} \end{align}\) .